A Recursive Method for Computing Zeta Functions of Varieties

Abstract

We present an algorithm that reduces the problem of calculating a numerical approximation to the action of absolute Frobenius on the middle-dimensional rigid cohomology of a smooth projective variety over a finite held, to that of performing the same calculation for a smooth hyperplane section. When combined with standard geometric techniques, this yields a method for computing zeta functions which proceeds ‘by induction on the dimension’. The ‘inductive step’ combines previous work of the author on the deformation of Frobenius with a higher rank generalisation of Kedlaya's algorithm. The analysis of the loss of precision during the algorithm uses a deep theorem of Christol and Dwork on p -adic solutions to differential systems at regular singular points. We apply our algorithm to compute the zeta functions of compactifications of certain surfaces which are double covers of the affine plane.